On weighted arcs with three non-zero characters (Q1331268)

A weighted arc in \({\mathbf P}{\mathbf G}(2,q)\) is a function \(f\) assigning to each point of \({\mathbf P}{\mathbf G}(2,q)\) a nonnegative integer such that the set of points assigned 0 to, contains \(k\) points and \(k < q^ 2 + q + 1\). For a point \(x\) the number \(f(x)\) is called the weight of \(x\) and for a line \(L\) the sum of the weights of the points on \(L\) is called the weight of \(L\). The weights of the lines of \({\mathbf P}{\mathbf G}(2,q)\) are the characters and the set of characters is the type of the arc. If \(n\) is the maximum character, then the weighted arc is more precisely called a \((k,n;f)\)-arc. The paper under review studies \((k,n;f)\)-arcs with only three different characters among which 0 (and trivially \(n\)). If the character different from 0 and \(n\) is \(n - 1\), then it is shown that \(f\) takes exactly two non-zero values \(w - 1\) and \(w\); the set of points with non-zero weight forms a maximal \((k,\rho)\)-arc in \({\mathbf P}{\mathbf G}(2,q)\) (i.e. every line has either 0 or \(\rho \neq 0\) points with non-zero weight) and there is a unique point with weight either \(w\) or \(w - 1\). If the character different from 0 and \(n\) is \(n - 2\), then the author obtains partial classification results.